Improving convergence of summations in heat conduction

James V. Beck, Kevin D. Cole

Research output: Contribution to journalArticlepeer-review

24 Scopus citations


This paper addresses exact, transient heat-conduction solutions in two-dimensional rectangles heated at a boundary. The standard method of separation of variables (SOV) solution has two parts, steady-state (or quasi-steady) and complementary transient. The steady-state component frequently converges slowly at the heated surface, which is usually the one of greatest interest. New procedures are given to construct a steady solution in the form of a single summation, one having eigenvalues in the homogeneous direction (yielding the same result as the standard SOV solution) and the other having eigenvalues in the non-homogeneous direction (called the non-standard solution). The non-standard solutions have much better convergence behavior at and near the heated boundary than the standard forms. Examples are given.

Original languageEnglish (US)
Pages (from-to)257-268
Number of pages12
JournalInternational Journal of Heat and Mass Transfer
Issue number1-2
StatePublished - Jan 2007


  • Analytical heat conduction
  • Convergence
  • Separation of variables
  • Series identities

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanical Engineering
  • Fluid Flow and Transfer Processes


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